L(s) = 1 | + (1.22 − 0.710i)2-s + (0.707 + 0.707i)3-s + (0.991 − 1.73i)4-s + (0.607 − 2.15i)5-s + (1.36 + 0.362i)6-s − 2.25·7-s + (−0.0216 − 2.82i)8-s + 1.00i·9-s + (−0.785 − 3.06i)10-s + (−1.66 − 1.66i)11-s + (1.92 − 0.527i)12-s + (4.76 + 4.76i)13-s + (−2.75 + 1.60i)14-s + (1.95 − 1.09i)15-s + (−2.03 − 3.44i)16-s + 6.99i·17-s + ⋯ |
L(s) = 1 | + (0.864 − 0.502i)2-s + (0.408 + 0.408i)3-s + (0.495 − 0.868i)4-s + (0.271 − 0.962i)5-s + (0.558 + 0.148i)6-s − 0.851·7-s + (−0.00765 − 0.999i)8-s + 0.333i·9-s + (−0.248 − 0.968i)10-s + (−0.500 − 0.500i)11-s + (0.556 − 0.152i)12-s + (1.32 + 1.32i)13-s + (−0.736 + 0.427i)14-s + (0.503 − 0.281i)15-s + (−0.508 − 0.860i)16-s + 1.69i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.91596 - 0.948607i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.91596 - 0.948607i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 + 0.710i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.607 + 2.15i)T \) |
good | 7 | \( 1 + 2.25T + 7T^{2} \) |
| 11 | \( 1 + (1.66 + 1.66i)T + 11iT^{2} \) |
| 13 | \( 1 + (-4.76 - 4.76i)T + 13iT^{2} \) |
| 17 | \( 1 - 6.99iT - 17T^{2} \) |
| 19 | \( 1 + (2.66 - 2.66i)T - 19iT^{2} \) |
| 23 | \( 1 - 4.41T + 23T^{2} \) |
| 29 | \( 1 + (-2.59 + 2.59i)T - 29iT^{2} \) |
| 31 | \( 1 + 3.93T + 31T^{2} \) |
| 37 | \( 1 + (-2.01 + 2.01i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.50iT - 41T^{2} \) |
| 43 | \( 1 + (7.14 - 7.14i)T - 43iT^{2} \) |
| 47 | \( 1 + 10.1iT - 47T^{2} \) |
| 53 | \( 1 + (0.649 - 0.649i)T - 53iT^{2} \) |
| 59 | \( 1 + (-5.64 - 5.64i)T + 59iT^{2} \) |
| 61 | \( 1 + (5.00 - 5.00i)T - 61iT^{2} \) |
| 67 | \( 1 + (4.95 + 4.95i)T + 67iT^{2} \) |
| 71 | \( 1 + 2.33iT - 71T^{2} \) |
| 73 | \( 1 - 2.18T + 73T^{2} \) |
| 79 | \( 1 - 6.38T + 79T^{2} \) |
| 83 | \( 1 + (-5.25 - 5.25i)T + 83iT^{2} \) |
| 89 | \( 1 + 15.7iT - 89T^{2} \) |
| 97 | \( 1 + 4.61iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.20114394223784564552653411418, −11.02932127117143942287895015746, −10.21233677666840746761978033410, −9.165086407989721668501255780227, −8.384851542490596636641585592257, −6.48901753846050436148970260757, −5.72393693489499974148610746947, −4.32436504341423367260712996923, −3.51243303274285753886472609696, −1.76040209392227235255643333080,
2.72202334598460095466742234201, 3.35643328473112885445282635538, 5.15550467279993417652683512438, 6.36697369893804993054936243245, 7.02413264500021888177132465955, 7.992474463452398623935093822980, 9.261811153948597184607472604515, 10.54853914168492150813817864156, 11.40565672242545273937899526271, 12.74737190269052831607027506725